Trilinear couplings are proportional to one of the vacuum expectation values \(v_u\) , \(v_d\)  and \(s\) , or to one of the trilinear soft supersymmetry breaking terms \(A_\lambda \)  or \(A_\kappa \) . The latter contributes only to the trilinear singlet Higgs couplings which play a negligible role for Higgs-to-Higgs decays since pure singlets have tiny production cross sections.The cross sections in Section 3 are derived including all couplings and dominant radiative corrections as included in NMSSMTools [1]}, [2]}, [3]}.
General expressions for the trilinear couplings can be found in
[4]}, but it is instructive to compare the ones relevant for Higgs-to-Higgs decays for the typical case \(s, A_\lambda \gg v_u, v_d \approx M_Z\) . In the Higgs basis these are (neglecting contributions of \({\cal O}(M_Z)\) )
\(a)&\sim \widehat{H} \widehat{H}_{SM} \widehat{H}_S: &\ &-\frac{\lambda }{\sqrt{2}} (2\kappa s + A_\lambda )\; , \nonumber \\b)&\sim \widehat{H} \widehat{H}_{SM} \widehat{H}_{SM}: &\ &0\; , \nonumber \\c)&\sim \widehat{H}_{S} \widehat{H}_{SM} \widehat{H}_{SM}: &\ &\frac{\lambda ^2}{\sqrt{2}} s - \frac{\lambda }{\sqrt{2}}\sin \beta \cos \beta (2\kappa s + A_\lambda )\; , \nonumber \\d)&\sim \widehat{H}_{S} \widehat{H} \widehat{H}: &\ &\frac{\lambda ^2}{\sqrt{2}} s + \frac{\lambda }{\sqrt{2}}\sin \beta \cos \beta (2\kappa s + A_\lambda )\; , \nonumber \\e)&\sim \widehat{A} \widehat{H}_{SM} \widehat{A}_S: &\ &\lambda {\sqrt{2}}\sin \beta \cos \beta (-2\kappa s + A_\lambda )\; , \nonumber \\f)&\sim \widehat{A} \widehat{H} \widehat{A}_S: &\ &\frac{\lambda }{\sqrt{2}}(\sin ^2\beta - \cos ^2\beta ) (-2\kappa s + A_\lambda )\; .\) 
